GMAT Math : Distances for absolute values

Absolute value data sufficiency questions are known to be notoriously hard but can be solved easily by thinking of absolute values as distances on the number line

Here is an illustrative example :

If b < a, is |x – a| < |x – b|

  1. 2x > a + b
  2. x > a

Traditionally, we would start by stress testing with actual numeric values for a, x and b – but this approach is a time sink and is prone to errors.

Behold! the distances paradigm to solve such a question :

Think of |x – a| as the distance between x and a on the number line. This sort of makes sense  – if we have the integers -3 and 2, then the distance between them would be |-3 – 2| = 5. If we actually plot -3 and 2 on the number line, we will observe that the distance between them is actually 5.

So if we agree with the above notion, we can re-frame the question into something simpler :Is the distance between x and a less than the distance between x and b

Given, b < a, so on the number line they would appear as :

b………………a

For the distance of x and a to be less than the distance between x and b, x will have to fall on any point after the midpoint of b and a.

b…….mid……x…a

or

b…….mid……a……..x

Let’s check the answer statements

Statement (1) tells us that 2x > a + b  or x > (a+b)/2. We recognize that (a+b)/2 is the average or the ‘mid point’ of a and b. The statement tells us that x is greater than the mid point. So information is sufficient.

Statement (2) tells us that x is greater than a. if we draw this out –

b……mid……a…..x

So information is sufficient again. We go with option (D) – each statement has sufficient information. Notice how we could rapidly move through this question by thinking of absolutes as distances. In our Hackbook we show how you can solve even the most complex absolute value questions using the distances property